Simplify The Expression:$\[ \frac{x^2-64}{7x} \cdot \frac{3x^3}{x^2+x-72} \\]

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including factoring, canceling, and combining like terms. In this article, we will simplify the given expression: ${ \frac{x^2-64}{7x} \cdot \frac{3x3}{x2+x-72} }$. We will break down the solution into manageable steps, making it easier for readers to follow along and understand the process.

Step 1: Factor the Numerators and Denominators

To simplify the given expression, we need to factor the numerators and denominators of both fractions. Let's start with the first fraction: ${ \frac{x^2-64}{7x} }$. The numerator can be factored as a difference of squares: x2−64=(x+8)(x−8)x^2 - 64 = (x + 8)(x - 8). The denominator is already factored as 7x7x.

# Factored Form of the First Fraction
## Numerator: (x + 8)(x - 8)
## Denominator: 7x

Now, let's move on to the second fraction: ${ \frac{3x3}{x2+x-72} }$. The numerator is already factored as 3x33x^3. The denominator can be factored as a difference of squares: x2+x−72=(x+9)(x−8)x^2 + x - 72 = (x + 9)(x - 8).

# Factored Form of the Second Fraction
## Numerator: 3x^3
## Denominator: (x + 9)(x - 8)

Step 2: Cancel Common Factors

Now that we have factored the numerators and denominators, we can cancel common factors between the two fractions. We can cancel the (x−8)(x - 8) term in the numerator of the first fraction with the (x−8)(x - 8) term in the denominator of the second fraction.

# Cancel Common Factors
## Cancel (x - 8) term
## Expression becomes: (x + 8) / (7x) * 3x^3 / (x + 9)

Step 3: Simplify the Expression

Now that we have canceled the common factors, we can simplify the expression further. We can multiply the numerators and denominators of the two fractions together.

# Simplified Expression
## Numerator: (x + 8) * 3x^3
## Denominator: 7x * (x + 9)

Step 4: Combine Like Terms

The final step is to combine like terms in the numerator and denominator. In the numerator, we have (x+8)∗3x3=3x4+24x3(x + 8) * 3x^3 = 3x^4 + 24x^3. In the denominator, we have 7x∗(x+9)=7x2+63x7x * (x + 9) = 7x^2 + 63x.

# Combined Like Terms
## Numerator: 3x^4 + 24x^3
## Denominator: 7x^2 + 63x

Step 5: Final Simplification

The final step is to simplify the expression further by canceling any common factors between the numerator and denominator. In this case, we can cancel the xx term in the numerator with the xx term in the denominator.

# Final Simplification
## Numerator: 3x^4 + 24x^3
## Denominator: 7x^2 + 63x
## Cancel x term
## Expression becomes: (3x^4 + 24x^3) / (7x^2 + 63x)

Conclusion

In this article, we simplified the given expression: ${ \frac{x^2-64}{7x} \cdot \frac{3x3}{x2+x-72} }$. We broke down the solution into manageable steps, making it easier for readers to follow along and understand the process. We factored the numerators and denominators, canceled common factors, simplified the expression, combined like terms, and finally simplified the expression further by canceling any common factors between the numerator and denominator.

Final Answer

The final simplified expression is: ${ \frac{3x^4 + 24x3}{7x2 + 63x} }$.

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Introduction

In our previous article, we simplified the given expression: ${ \frac{x^2-64}{7x} \cdot \frac{3x3}{x2+x-72} }$. We broke down the solution into manageable steps, making it easier for readers to follow along and understand the process. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to factor the numerators and denominators. This involves breaking down the expression into simpler components, such as differences of squares or cubes.

Q: How do I factor a difference of squares?

A: To factor a difference of squares, you can use the formula: a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b). For example, x2−64=(x+8)(x−8)x^2 - 64 = (x + 8)(x - 8).

Q: What is the next step after factoring the numerators and denominators?

A: After factoring the numerators and denominators, the next step is to cancel common factors between the two fractions. This involves identifying any common terms in the numerator and denominator and canceling them out.

Q: How do I simplify an expression with multiple fractions?

A: To simplify an expression with multiple fractions, you can multiply the numerators and denominators of each fraction together. Then, you can combine like terms and cancel any common factors.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to combine like terms and cancel any common factors. This involves simplifying the expression as much as possible to make it easier to work with.

Q: Can I use a calculator to simplify an algebraic expression?

A: While a calculator can be a useful tool for simplifying algebraic expressions, it is not always the best option. Calculators can be prone to errors, and they may not always provide the simplest form of the expression. It is often better to simplify expressions by hand to ensure accuracy and to understand the underlying math.

Q: How do I know when an expression is fully simplified?

A: An expression is fully simplified when there are no more like terms to combine and no more common factors to cancel. You can check by looking for any remaining terms that can be combined or canceled.

Example Problems

Problem 1: Simplify the expression ${

\frac{x^2-49}{x+7} \cdot \frac{x^2+7x}{x-7} }$.

Solution: To simplify this expression, we can factor the numerators and denominators, cancel common factors, and combine like terms.

# Factored Form of the First Fraction
## Numerator: (x + 7)(x - 7)
## Denominator: x + 7

# Factored Form of the Second Fraction
## Numerator: x(x + 7)
## Denominator: x - 7

# Cancel Common Factors
## Cancel (x + 7) term
## Expression becomes: (x - 7) / (x - 7) * x(x + 7) / 1

# Simplified Expression
## Numerator: x(x + 7)
## Denominator: 1

Problem 2: Simplify the expression ${

\frac{x^2+10x+25}{x+5} \cdot \frac{x^2-10x+25}{x-5} }$.

Solution: To simplify this expression, we can factor the numerators and denominators, cancel common factors, and combine like terms.

# Factored Form of the First Fraction
## Numerator: (x + 5)(x + 5)
## Denominator: x + 5

# Factored Form of the Second Fraction
## Numerator: (x - 5)(x - 5)
## Denominator: x - 5

# Cancel Common Factors
## Cancel (x + 5) term
## Expression becomes: (x + 5) / (x + 5) * (x - 5)(x - 5) / 1

# Simplified Expression
## Numerator: (x - 5)(x - 5)
## Denominator: 1

Conclusion

In this article, we answered some frequently asked questions related to simplifying algebraic expressions. We covered topics such as factoring, canceling common factors, and combining like terms. We also provided example problems to help illustrate the concepts. By following these steps and practicing with example problems, you can become more confident in your ability to simplify algebraic expressions.

Final Answer

The final simplified expression is: ${ \frac{3x^4 + 24x3}{7x2 + 63x} }$.